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# Archard equation

The Archard equation is a simple model used to describe sliding wear and is based around the theory of asperity contact.

## Equation $Q = \frac {KW}H$

where:

Q is the total volume of wear debris produced per unit distance moved
W is the total normal load
H is the hardness
K is a dimensionless constant

## Derivation

The equation can be derived by first examining the behavior of a single asperity.

The local load $\, \delta W$, supported by an asperity, assumed to have a circular cross-section with a radius $\, a$, is: $\delta W = P \pi {a^2} \,\!$

where P is the yield pressure for the asperity, assumed to be deforming plastically. P will be close to the indentation hardness, H, of the asperity.

If the volume of wear debris, $\, \delta V$, for a particular asperity is a hemisphere sheared off from the asperity, it follows that: $\delta V = \frac 2 3 \pi a^3$

This fragment is formed by the material having slid a distance 2a

Hence, $\, \delta Q$, the wear volume of material produced from this asperity per unit distance moved is: $\delta Q = \frac {\delta V} {2a} = \frac {\pi a^2} 3 \equiv \frac {\delta W} {3P} \approx \frac {\delta W} {3H}$ making the approximation that $\,P \approx H$

However, not all asperities will have had material removed when sliding distance 2a. Therefore, the total wear debris produced per unit distance moved, $\, Q$ will be lower than the ratio of W to 3H. This is accounted for by the addition of a dimensionless constant K, which also incorporates the factor 3 above. These operations produce the Archard equation as given above.

K is therefore a measure of the severity of wear. Typically for 'mild' wear, K ≈ 10−8, whereas for 'severe' wear, K ≈ 10−2.